ADVANCED TIME AVERAGE DIGITAL HOLOGRAPHY BY MEANS OF FREQUENCY AND ___PHASE MODULATION

ADVANCED TIME AVERAGE DIGITAL HOLOGRAPHY BY MEANS OF FREQUENCY AND ___PHASE MODULATION

PhD dissertation

Study program: P3901 Applied Sciences Engineering Field of study: 3901V055 Applied Sciences Engineering

Author: Ing. Pavel Psota

Supervisor: prof. Ing. Václav Kopecký, CSc.

Tutor: Ing. Vít Lédl, Ph.D.

Liberec 2015

2.7 × 10 ⁻⁴ 𝜆





𝑬(𝒓, 𝑡)

𝑯(𝒓, 𝑡) 𝒓 𝑡

𝛻 × 𝑯 = 𝜀 ₀ 𝜕𝑬

𝜕𝑡 ,

𝛻 × 𝑬 = −𝜇 ₀ 𝜕𝑯

𝜕𝑡 ,

𝛻. 𝑬 = 0,

𝛻. 𝑯 = 0,

(𝛻 ×)

(𝛻. ) 𝑬, 𝑯

𝜇 ₀ = 4𝜋 × 10 ⁻⁷ 𝐻𝑚 ⁻¹ 𝜀 ₀ = 8,85 × 10 ⁻¹² 𝐹𝑚 ⁻¹

𝑫 𝑩

𝑫 𝑬

𝑩 𝑯

𝑫 = 𝜀 ₀ 𝑬 + 𝑷,

𝑩 = 𝜇 ₀ 𝑯 + 𝑴

𝑷 𝑴

𝑬 𝑯 𝑫 𝑩

𝛻 × 𝑯 = 𝜀 ₀ 𝜕𝑫

𝜕𝑡 ,

𝛻 × 𝑬 = −𝜇 ₀ 𝜕𝑩

𝜕𝑡 ,

𝛻. 𝑫 = 0,

𝛻. 𝑩 = 0.

𝑷 𝑬

𝑷 = 𝑴 = 𝟎

𝛻 × (𝛻 × 𝑬) = 𝛻. (𝛻. 𝑬) − 𝛻 ² 𝑬

𝛻 ² 𝑬(𝒓, 𝑡) − 1 𝑐 ²

𝜕 ² 𝑬(𝒓, 𝒕)

𝜕𝑡 ² = 0,

𝛻 ² 𝑯(𝒓, 𝑡) − 1 𝑐 ²

𝜕 ² 𝑯(𝒓, 𝑡)

𝜕𝑡 ² = 0, 𝑐 = √ _𝜇 ¹

0 𝜀 0 = 299 792 458 𝑚𝑠 ⁻¹ 𝛻 ² = ^{𝜕 2}

𝜕𝑥 ² + ^{𝜕 2}

𝜕𝑦 ² +

𝜕 ²

𝜕𝑧 ²

𝑬 𝑯

𝛻 ² 𝑢(𝒓, 𝑡) − 1 𝑐 ²

𝜕 ² 𝑢(𝒓, 𝑡)

𝜕𝑡 ² = 0, 𝑢(𝒓, 𝑡)

𝑺

𝑺 = 𝑬 × 𝑯

𝐼

𝐼 = 〈|𝑺|〉

𝑬 𝑯

𝜔

𝑬(𝒓, 𝑡) = 𝑬 _𝟎 (𝒓)𝑐𝑜𝑠(𝜔𝑡 + 𝜑(𝒓)),

𝑯(𝒓, 𝑡) = 𝑯 _𝟎 (𝒓)𝑐𝑜𝑠(𝜔𝑡 + 𝜑(𝒓)),

𝑬 𝟎 (𝒓) 𝑯 𝟎 (𝒓) 𝜑(𝒓)

𝒓 𝜔𝑡

𝑗 = √−1

1

𝑬(𝒓, 𝑡) = 𝑅𝑒{𝑬̇(𝒓)𝑒𝑥𝑝(𝑗𝜔𝑡)},

𝑯(𝒓, 𝑡) = 𝑅𝑒{𝑯̇(𝒓)𝑒𝑥𝑝(𝑗𝜔𝑡)},

𝑬̇(𝒓) 𝑯̇(𝒓)

𝜔

𝑓 𝑇

𝜔 = 2𝜋𝑓 = 2𝜋 𝑇 .

𝑬̇ 𝑯̇ 𝑑/𝑑𝑡 = 𝑗𝜔

𝛻 × 𝑯̇ = 𝑗𝜀 ₀ 𝜔𝑬̇,

𝛻 × 𝑬̇ = −𝑗𝜇 ₀ 𝜔𝑯̇,

𝛻. 𝑬̇ = 0,

𝛻. 𝑯̇ = 0,

𝛻 ² 𝑈(𝒓) + 𝑘 ² 𝑈(𝒓) = 0,

𝑈(𝒓) 𝑬̇(𝒓)

𝑯̇(𝒓) 𝑘 = 𝜔/𝑐

𝑈(𝑡) 𝑈(𝑟)

𝑈(𝑡) ω

〈𝑺〉 = 〈𝑅𝑒{𝑬̇(𝒓)𝑒𝑥𝑝(𝑗𝜔𝑡)} × 𝑅𝑒{𝑯̇(𝒓)𝑒𝑥𝑝(𝑗𝜔𝑡)}〉 = 𝑅𝑒{𝑺̇},

𝑺̇

𝑺̇ = 1

2 𝑬̇ × 𝑯̇ ^∗ ,

𝐼 = |𝑅𝑒{𝑺̇}|

𝑬̇(𝒓) 𝑯̇(𝒓)

𝒌

𝑬̇(𝒓) = 𝑬 _𝟎 𝑒𝑥𝑝(−𝑗𝒌𝒓),

𝑯̇(𝒓) = 𝑯 _𝟎 𝑒𝑥𝑝(−𝑗𝒌𝒓),

𝑬 _𝟎 𝑯 _𝟎

𝑬̇(𝒓) 𝑯̇(𝒓)

𝒌 × 𝑯 _𝟎 = −𝜔𝜀 ₀ 𝑬 _𝟎 ,

𝒌 × 𝑬 _𝟎 = 𝜔𝜇 ₀ 𝑯 _𝟎 .

𝑬 𝑯

𝑯 𝑬

𝒌 𝑬 𝑯

𝑬 𝑯 𝒌

𝜆

𝜆 = 𝑐𝑇 = 𝑐 𝑓 = 2𝜋

𝑘 .

human eye. VIS is

𝑟 = 𝑟 ₀

𝒌 = 𝑘𝒆 _𝟑 ,

𝒆 𝟏 , 𝒆 𝟐 , 𝒆 𝟑

𝑬 = 𝐸 ₀ 𝑐𝑜𝑠(𝜔𝑡 − 𝑘𝑧)𝒆 _𝟏 ,

𝑯 = 𝐻 ₀ 𝑐𝑜𝑠(𝜔𝑡 − 𝑘𝑧)𝒆 _𝟐 .

𝑬

+ 𝜋 2 ⁄

𝑬 = 𝐸 ₀ 𝑐𝑜𝑠 (𝜔𝑡 − 𝑘𝑧 + 𝜋

2 ) 𝒆 _𝟐 = 𝐸 ₀ 𝑠𝑖𝑛(𝜔𝑡 − 𝑘𝑧)𝒆 _𝟐 , 𝑯 = 𝐻 ₀ 𝑐𝑜𝑠 (𝜔𝑡 − 𝑘𝑧 + 𝜋

2 ) 𝒆 _𝟏 = 𝐻 ₀ 𝑠𝑖𝑛(𝜔𝑡 − 𝑘𝑧)𝒆 _𝟏 .

𝑬 = 𝐸 ₀ [𝑐𝑜𝑠(𝜔𝑡 − 𝑘𝑧)𝒆 _𝟏 − 𝑠𝑖𝑛(𝜔𝑡 − 𝑘𝑧)𝒆 _𝟐 ],

𝑯 = 𝐻 ₀ [𝑐𝑜𝑠(𝜔𝑡 − 𝑘𝑧)𝒆 _𝟏 + 𝑠𝑖𝑛(𝜔𝑡 − 𝑘𝑧)𝒆 _𝟐 ].

𝑬 _𝒙 = 𝐸 ₀ 𝑐𝑜𝑠(𝜔𝑡) 𝑎𝑛𝑑 𝑬 _𝒚 = −𝐸 ₀ 𝑠𝑖𝑛(𝜔𝑡),

𝑯 _𝒙 = 𝐻 ₀ 𝑠𝑖𝑛(𝜔𝑡) 𝑎𝑛𝑑 𝑯 _𝒚 = 𝐻 ₀ 𝑐𝑜𝑠(𝜔𝑡),

𝑬 ^𝟐 = 𝑬 _𝒙 ^𝟐 + 𝑬 _𝒚 ^𝟐

𝑯 ^𝟐 = 𝑯 _𝒙 ^𝟐 + 𝑯 _𝒚 ^𝟐 ,

𝑬 _𝒙 = 𝐸 ₀₁ 𝑐𝑜𝑠(𝜔𝑡 − 𝜃 ₁ ) 𝑎𝑛𝑑 𝑬 _𝒚 = −𝐸 ₀₂ 𝑐𝑜𝑠(𝜔𝑡 − 𝜃 ₂ ).

( 𝑬 _𝒙 𝐸 ₀₁ )

𝟐

+ ( 𝑬 _𝒚 𝐸 ₀₂ )

𝟐

− 2 𝑬 _𝒙 𝐸 ₀₁

𝑬 _𝒚

𝐸 ₀₂ 𝑐𝑜𝑠(𝜃 ₂ − 𝜃 ₁ ) = 𝑠𝑖𝑛 ² (𝜃 ₂ − 𝜃 ₁ ).

𝜃 2 − 𝜃 1 = ^𝜋

2 , ^3𝜋

2 , …

𝐸 ₀₁ = 𝐸 ₀₂

𝜃 2 − 𝜃 1 = 0, 𝜋, 2𝜋, …

𝑬 _𝒙 = ∓ 𝐸 ₀₂ 𝐸 ₀₁ 𝑬 _𝒚 ,

𝑈 ₁ (𝒓), 𝑈 ₂ (𝒓),

𝑈(𝒓) = 𝑈 ₁ (𝒓) + 𝑈 ₂ (𝒓).

𝐼 ₁ = |𝑈 ₁ | ² 𝐼 ₂ = |𝑈 ₂ | ²

𝐼(𝒓) = |𝑈(𝒓)| ² = |𝑈 ₁ (𝒓) + 𝑈 ₂ (𝒓)| ² =

= |𝑈 ₁ (𝒓)| ² + |𝑈 ₂ (𝒓)| ² + 𝑈 ₁ (𝒓)𝑈 ₂ ^∗ (𝒓) + 𝑈 ₁ ^∗ (𝒓)𝑈 ₂ (𝒓).

𝑈 ₁ = √𝐼 1 𝑒 ^𝑗𝜑

𝑈 ₂ = √𝐼 2 𝑒 ^𝑗𝜑

, 𝜑 ₁ , 𝜑 ₂

𝐼 = 𝐼 ₁ + 𝐼 ₂ + 2√𝐼 ₁ 𝐼 ₂ 𝑐𝑜𝑠𝜑,

𝜑 = 𝜑 2 − 𝜑 1

𝑈 𝜑

𝑈 ₁ , 𝑈 ₂

𝐼

𝜑 𝑈 ₁ , 𝑈 ₂

𝜑 = 𝜑 2 − 𝜑 1

2√𝐼 1 𝐼 2 𝑐𝑜𝑠𝜑

𝑉 = 𝐼 _𝑚𝑎𝑥 − 𝐼 _𝑚𝑖𝑛 𝐼 _𝑚𝑎𝑥 + 𝐼 _𝑚𝑖𝑛

𝐼 𝑚𝑎𝑥 𝐼 𝑚𝑖𝑛

𝐼 _𝑚𝑎𝑥 𝐼 _𝑚𝑖𝑛 𝜑 = 0 𝜑 = 𝜋

𝑉 = 2√𝐼 ₁ 𝐼 ₂ 𝐼 ₁ + 𝐼 ₂ .

𝐼 ₁ = 𝐼 ₂

𝑉 = 1

𝐺(𝜏) = 〈𝑈 ^∗ (𝑡)𝑈(𝑡 + 𝜏)〉 = 𝑙𝑖𝑚 1

𝑇→∞ 2𝑇

∫ 𝑈 ^∗ (𝑡)𝑈(𝑡 + 𝜏)𝑑𝑡.

𝑇

−𝑇

𝐼 = 𝐺(0) 𝐺(𝜏) 𝜏 = 0

𝐺(𝜏) 𝐼 = 𝐺(0)

𝑔(𝜏) = 𝐺(𝜏)

𝐺(0) = 〈𝑈 ^∗ (𝑡)𝑈(𝑡 + 𝜏)〉

〈𝑈 ^∗ (𝑡)𝑈(𝑡)〉 ,

0 ≤ |𝑔(𝜏)| ≤ 1.

𝐼 = 𝐼 ₁ + 𝐼 ₂ + 2√𝐼 ₁ 𝐼 ₂ |𝑔(𝜏)|𝑐𝑜𝑠𝜑

𝑉 = ^{2√𝐼 1 𝐼 2}

𝐼 ₁ +𝐼 ₂ |𝑔(𝜏)|.

𝐼 ₁ = 𝐼 ₂

𝑉 = |𝑔(𝜏)|

|𝑔(𝜏)| = 1

|𝑔(𝜏)| = 0

0 < |𝑔(𝜏)| < 1

𝑟 ₁ , 𝑟 ₂

𝐺(𝒓 _𝟏 , 𝒓 _𝟐 , 𝜏) = 〈𝑈 ^∗ (𝒓 _𝟐 , 𝑡)𝑈(𝒓 _𝟏 , 𝑡 + 𝜏)〉 =

= 𝑙𝑖𝑚 1

𝑇→∞ 2𝑇

∫ 𝑈 ^∗ (𝒓 _𝟐 , 𝑡)𝑈(𝒓 _𝟏 , 𝑡 + 𝜏)𝑑𝑡

𝑇

−𝑇

,

𝑔(𝒓 _𝟏 , 𝒓 _𝟐 𝜏) = 𝐺(𝒓 _𝟏 , 𝒓 _𝟐 , 𝜏)

√𝐺(𝒓 _𝟏 , 𝒓 _𝟏 , 0)𝐺(𝒓 _𝟐 , 𝒓 _𝟐 , 0) ,

𝐺(𝒓 𝟏 , 𝒓 𝟏 , 0) 𝒓 𝟏 𝐺(𝒓 𝟐 , 𝒓 𝟐 , 0) 𝒓 𝟐

𝒓 𝟏 𝑡 𝒓 𝟐 𝑡 + 𝜏





𝜀(𝒓)





𝑈(𝑃)

𝑃

𝑃

𝑈(𝑃) 𝑃

𝑘 𝑈 0

𝑈(𝑃) = 𝑗𝑈 ₀

2𝜆 ∬ 𝑒𝑥𝑝 [−𝑗𝑘(𝑟 ₀ + 𝑟 ₁ )]

𝑟 ₀ 𝑟 ₁

𝑆1

(𝑐𝑜𝑠(𝑟 ₁ , 𝒏) + 𝑐𝑜𝑠 (𝑟 ₀ , 𝒏))𝑑𝑆.

𝑗/2𝜆 (𝑐𝑜𝑠(𝑟 ₁ , 𝒏) + 𝑐𝑜𝑠 (𝑟 ₀ , 𝒏)) 𝒏

𝑈

𝑈

𝑐𝑜𝑠(𝑟 ₀ , 𝒏) = 1

𝑐𝑜𝑠(𝑟 ₁ , 𝒏)

𝑗/𝜆

𝑈(𝑃) = 𝑗𝑈 ₀

𝜆 ∬ 𝑒𝑥𝑝 [−𝑗𝑘(𝑟 ₀ + 𝑟 ₁ )]

𝑟 ₀ 𝑟 ₁

𝑆1

𝑑𝑆,

(𝑥 0 , 𝑦 0 )

𝑃 0 (𝑥 0 , 𝑦 0 , 0) 𝑃 1 (𝑥, 𝑦, 𝑧)

|𝑃 ₀ 𝑃 1 | 𝑟

𝑈(𝑃 ₁ )

𝑈(𝑃 ₁ ) = 𝑗

𝜆 ∬ ℎ(𝑥 ₀ , 𝑦 ₀ , 0) 𝑒𝑥𝑝 [−𝑗𝑘𝑟]

𝑆1 𝑟

𝑑𝑥 ₀ 𝑑𝑦 ₀ ,

𝑟 = √(𝑥 ₀ − 𝑥) ² + (𝑦 ₀ − 𝑦) ² + 𝑧 ² . 𝑟

1 ℎ(𝑥

, 𝑦

, 0) ℎ(𝑥

, 𝑦

, 0) = 1 ℎ(𝑥

, 𝑦

, 0) ∈ 𝑆1

ℎ(𝑥

, 𝑦

, 0) = 0 ℎ(𝑥

, 𝑦

, 0)

𝑟 = √(𝑥 ₀ − 𝑥) ² + (𝑦 ₀ − 𝑦) ² + 𝑧 ² ~𝑧 + 1

2𝑧 [(𝑥 ₀ − 𝑥) ² + (𝑦 ₀ − 𝑦) ² ]

− 1

8𝑧 [(𝑥 ₀ − 𝑥) ² + (𝑦 ₀ − 𝑦) ² ] ² + ⋯

𝑈(𝑥, 𝑦, 𝑧) = 𝑗

𝑧𝜆 𝑒𝑥𝑝 (− 𝑗𝜋

𝜆𝑧 [𝑥 ² + 𝑦 ² ]) × × ∬ ℎ(𝑥 ₀ , 𝑦 ₀ , 0) 𝑒𝑥𝑝 (− 𝑗𝜋

𝜆𝑧 [𝑥 ₀ ² + 𝑦 ₀ ² ])

∞

−∞

×

× 𝑒𝑥𝑝 (−𝑗2𝜋[𝑥 ₀ 𝑥

𝜆𝑧 + 𝑦 ₀ 𝑦

𝜆𝑧 ]) 𝑑𝑥 ₀ 𝑑𝑦 ₀ .

1

8 𝑘𝑧 ( 𝑥 ² + 𝑦 ² 𝑧 ² )

2

≪ 𝜋 2 .

𝜋/2

𝑈(𝑥, 𝑦, 𝑧) ( ^𝑥

^+𝑦

𝑧

) = 𝑡𝑎𝑛 ² (𝑟 ₁ , 𝒏)

𝑡𝑎𝑛 ⁴ (𝑟 ₁ , 𝒏) < 2𝜆 𝑧 .

𝑒𝑥𝑝 (− ^𝑗𝜋

𝜆𝑧 [𝑥 0 2 + 𝑦 ₀ ² ]) ≈ 1

𝑈(𝑥, 𝑦, 𝑧) = 𝑗

𝑧𝜆 𝑒𝑥𝑝 (− 𝑗𝜋

𝜆𝑧 [𝑥 ² + 𝑦 ² ]) ×

× ∬ 𝑒𝑥𝑝 (−𝑗2𝜋[𝑥 ₀ 𝑥

𝜆𝑧 + 𝑦 ₀ 𝑦

𝜆𝑧 ]) 𝑑𝑥 ₀ 𝑑𝑦 ₀

∞

−∞

^𝑗𝜋

𝜆𝑧 [𝑥 0 2 + 𝑦 0 2 ] ≪ ^𝜋

2

𝑈 0

𝑈 𝑟

ℎ

ℎ ≈ |𝑈 _𝑜 + 𝑈 _𝑟 | ² = |𝑈 _𝑜 | ² + |𝑈 _𝑟 | ² + 𝑈 _𝑜 𝑈 _𝑟 ^∗ + 𝑈 _𝑜 ^∗ 𝑈 _𝑟

ℎ

𝑈 _𝑜

1

𝑈 _𝑟

𝑈 = ℎ𝑈 _𝑟 ≈ 𝑈 _𝑟 (𝐼 _𝑟 + 𝐼 ₀ ) + 𝐼 _𝑟 𝑈 ₀ + 𝑈 _𝑟 ² 𝑈 ₀ ^∗ .

𝐼 _𝑟 𝑈 ₀ 𝐼 _𝑟 𝐼 _𝑟

𝑈 _𝑟 ² 𝑈 ₀ ^∗ 𝑈 _𝑟 ²

𝑈 𝑟 (𝐼 𝑟 + 𝐼 0 )

𝐼 𝑟 𝐼 0

𝜗

𝑈 ≈ 𝑈 _𝑟 (𝐼 _𝑟 + 𝐼 ₀ ) + 𝐼 _𝑟 𝑒𝑥𝑝(−𝑗𝑘𝑥 𝑠𝑖𝑛(𝜗)) 𝑈 ₀ + 𝑈 _𝑟 ² 𝑒𝑥𝑝(𝑗𝑘𝑥 𝑠𝑖𝑛(𝜗)) 𝑈 ₀ ^∗ . 𝑒𝑥𝑝(−𝑗𝑘𝑥 𝑠𝑖𝑛(𝜗))

−𝜗 𝑈 𝑟

𝜗 𝑈 _𝑟

θ

1/𝛬 𝜃

𝛬 = 𝜆 𝑠𝑖𝑛𝜃 .

Λ

𝛬 > 2∆𝜉,

∆𝜉 𝜃 𝑠𝑖𝑛𝜃 ≈ 𝜃

𝜃 _𝑚𝑎𝑥

𝜃 _𝑚𝑎𝑥 ≈ 𝜆

2∆𝜉 .

𝑑 ₀ 𝑥

𝑑 ₀ 𝑑

𝑡𝑎𝑛𝜃 = 𝑑 ₀

2 + 𝑁∆𝜉

2

𝑑 ,

𝑁 𝑥

𝜃 _𝑚𝑎𝑥

𝜆 2∆𝜉 =

𝑑 ₀ 2 +

𝑁∆𝜉 2 𝑑

𝑑 ₀ < 𝜆𝑑

∆𝜉 − 𝑁∆𝜉,

𝑑 ₀ 𝑑

1

tanθ ≈ θ

𝑑 𝑣 𝑑 0 1

𝑓 = ¹

𝑔 − ¹

𝑏 𝑍 = ^𝑑

𝑑

= ^𝑓

𝑔−𝑓

𝑡𝑎𝑛𝜃 = ^𝑑

2(𝑎+𝑏)

𝑎 = 𝑓𝑔

𝑔 − 𝑓 − 𝑓𝑑 ₀ (𝑔 − 𝑓)2𝑡𝑎𝑛𝜃 .

𝑑 = 𝑎 + 𝑏 𝑑 = 𝑎 + 𝑔 𝜃 _{𝑟𝑒𝑑𝑢𝑐𝑒𝑑} ≪

𝜃 𝑙𝑒𝑛𝑠𝑙𝑒𝑠𝑠

ℎ 𝑈𝑟 ^∗

𝜉, 𝜂

𝑑

𝑥, 𝑦

𝜃 1 𝜃 2

𝑈𝑟 ^∗ = 𝐴𝑒𝑥𝑝 𝑗2𝜋

𝜆 (𝑠𝑖𝑛𝜃 ₁ 𝜉 + 𝑠𝑖𝑛𝜃 ₂ 𝜂).

𝑑 𝑆𝑝ℎ

𝑈𝑟 ^∗ = 𝐴𝑒

𝑗2𝜋𝑑 𝑆𝑝ℎ

𝜆 𝑒 ^{𝜆𝑑 𝑗𝜋 (𝜉 2 +𝜂 2 )} .

𝑈 (𝑥, 𝑦)

𝑈(𝑥, 𝑦) = 1

𝑗𝜆 ∬ ℎ(𝜉, 𝜂)𝑈𝑟 ^∗ (𝜉, 𝜂) 𝑒𝑥𝑝(𝑗𝑘𝑟) 𝑟 𝑑𝜉𝑑𝜂,

𝑟 = √𝑑 ² + (𝜉 − 𝑥) ² + (𝜂 − 𝑦) ² = 𝑑√1 + ^{(𝜉−𝑥)}

^{+(𝜂−𝑦)}

𝑑

.

𝑟

𝑈(𝑥, 𝑦, 𝑑) = 𝑗

𝑑𝜆 𝑒𝑥𝑝 (− 𝑗𝜋

𝜆𝑑 [𝑥 ² + 𝑦 ² ]) ×

× ∬ ℎ(𝜉, 𝜂) 𝑈𝑟 ^∗ (𝜉, 𝜂)𝑒𝑥𝑝 (− 𝑗𝜋

𝜆𝑑 [𝜉 ² + 𝜂 ² ])

∞

−∞

×

× 𝑒𝑥𝑝 (−𝑗2𝜋[𝜉 𝑥

𝜆𝑑 + 𝜂 𝑦

𝜆𝑑 ]) 𝑑𝜉𝑑𝜂.

𝛥𝜉 × 𝛥𝜂 𝑁 × 𝑀

𝜉 = 𝑘𝛥𝜉 𝑤ℎ𝑒𝑟𝑒 1 < 𝑘 < 𝑁 𝑎𝑛𝑑 𝜂 = 𝑙𝛥𝜂 𝑤ℎ𝑒𝑟𝑒 1 < 𝑙 < 𝑀.

𝑥 = 𝑛𝛥𝑥 𝑤ℎ𝑒𝑟𝑒 1 < 𝑛 < 𝑁 𝑎𝑛𝑑 𝑦 = 𝑚𝛥𝑦 𝑤ℎ𝑒𝑟𝑒 1 < 𝑚 < 𝑀.

𝑈(𝑛𝛥𝑥, 𝑚𝛥𝑦) = 𝑗

𝑑𝜆 𝑒𝑥𝑝 (− 𝑗𝜋

𝜆𝑑 [(𝑛𝛥𝑥) ² + (𝑚𝛥𝑦) ² ]) ×

× ∑ ∑ ℎ(𝑘𝛥𝜉, 𝑙𝛥𝜂)

𝑀

𝑚=1 𝑁

𝑛=1

𝑈𝑟 ^∗ (𝑘𝛥𝜉, 𝑙𝛥𝜂) ×

× 𝑒𝑥𝑝 (− 𝑗𝜋

𝜆𝑑 [(𝑘𝛥𝜉) ² + (𝑙𝛥𝜂) ² ]) ×

× 𝑒𝑥𝑝 (−𝑗2𝜋 ( 𝑘𝑛 𝑁 + 𝑙𝑚

𝑀 )) 𝑑𝜉𝑑𝜂.

𝛥𝑥, 𝛥𝑦 𝑁∆𝜉 × 𝑀∆𝜂

𝑑 𝜆

∆𝑥 = 𝜆𝑑

𝑁∆𝜉 𝑎𝑛𝑑 ∆𝑦 = 𝜆𝑑 𝑀∆𝜂 .

𝑈(𝑛𝛥𝑥, 𝑚𝛥𝑦) = 𝑗

𝑑𝜆 𝑒𝑥𝑝 (− 𝑗𝜋

𝜆𝑑 [(𝑛𝛥𝑥) ² + (𝑚𝛥𝑦) ² ]) ×

× ℱ ⁻¹ {ℎ(𝑘∆𝜉, 𝑙∆𝜂)𝑈𝑟 ^∗ (𝑘∆𝜉, 𝑙∆𝜂)𝑒𝑥𝑝 (− 𝑗𝜋

𝜆𝑑 [(𝑘𝛥𝜉) ² + (𝑙𝛥𝜂) ² ])}, ℱ ⁻¹

𝑈(𝑥, 𝑦) = ∬ ℎ(𝜉, 𝜂)𝑈𝑟 ^∗ (𝜉, 𝜂)𝑔(𝑥 − 𝜉, 𝑦 − 𝜂) 𝑑𝜉𝑑𝜂

𝑔(𝑥, 𝑦) = 1 𝑗𝜆

𝑒𝑥𝑝(𝑗𝑘√𝑑 ² + 𝑥 ² + 𝑦 ² )

√𝑑 ² + 𝑥 ² + 𝑦 ² .

𝑔(𝑥, 𝑦, 𝜉, 𝜂) = 𝑔(𝑥 − 𝜉, 𝑦 − 𝜂)

𝑈(𝑥, 𝑦) = (ℎ𝑈𝑟 ^∗ ) ∗ 𝑔 = ℱ ⁻¹ {ℱ(ℎ𝑈𝑟 ^∗ )ℱ(𝑔)}.

𝑔(𝑛, 𝑚) =

= 1 𝑗𝜆

𝑒𝑥𝑝 (𝑗 2𝜋

𝜆 √ 𝑑 ² + (𝑛 − 𝑁/2) ² ∆𝜉 ² + (𝑚 − 𝑀/2) ² ∆𝜂 ² )

√𝑑 ² + (𝑛 − 𝑁/2) ² ∆𝜉 ² + (𝑚 − 𝑀/2) ² ∆𝜂 ² .

𝑁/2 𝑀/2

𝐺(𝑘, 𝑙) 𝑔(𝑛, 𝑚)

𝐺(𝑘, 𝑙)

𝐺(𝑘, 𝑙) = 𝑒𝑥𝑝 (

𝑗 2𝜋𝑑 𝜆

√ 1 −

𝜆 ² ( 𝑘 + 𝑁 ² ∆𝜉 ² 2𝑑𝜆 )

2

𝑁 ² ∆𝜉 ² +

𝜆 ² ( 𝑙 + 𝑀 ² ∆𝜂 ² 2𝑑𝜆 )

2

𝑀 ² ∆𝜂 ²

) ,

𝑈(𝑥, 𝑦) = ℱ ⁻¹ {ℱ(ℎ𝑈𝑟 ^∗ )𝐺}.

∆𝑥 = ∆𝜉 𝑎𝑛𝑑 ∆𝑦 = ∆𝜂.

𝛥𝜉 × 𝛥𝜂

𝑁∆𝜉 × 𝑀∆𝜂

𝑓 = (1/𝑑 + 1/𝑑̇) ⁻¹

𝑑̇ 𝑑̇ = 𝑑𝑀 𝑀

𝐿(𝜉, 𝜂) = 𝑒𝑥𝑝 [ 𝑗𝜋

𝜆 (1/𝑑 + 1/𝑑̇)(𝜉 ² + 𝜂 ² )]

𝑈(𝑥, 𝑦) = ℱ ⁻¹ {ℱ(𝐿 ℎ 𝑈𝑟 ^∗ )ℱ(𝑔)}.

𝑑̇ 𝑑

𝑀 = 1

∆𝑥 = ∆𝜉 ∆𝑦 = ∆𝜂

𝑈(𝑛∆𝑥, 𝑚∆𝑦)

𝐼(𝑛∆𝑥, 𝑚∆𝑦) 𝜑(𝑛∆𝑥, 𝑚∆𝑦)

𝐼(𝑛∆𝑥, 𝑚∆𝑦) = |𝑈(𝑛∆𝑥, 𝑚∆𝑦)| ² ,

𝜑(𝑛∆𝑥, 𝑚∆𝑦) = 𝑎𝑟𝑐𝑡𝑎𝑛 𝐼𝑚{𝑈(𝑛∆𝑥, 𝑚∆𝑦)}

𝑅𝑒{𝑈(𝑛∆𝑥, 𝑚∆𝑦)} ,

𝑑 = 0.6 𝑚

𝑁 = 𝑀 = 2048 𝑝𝑖𝑥 ∆𝜉 = ∆𝜂 = 3.45 𝜇𝑚 𝜆 = 532𝑛𝑚

𝑁∆𝑥 = ^𝜆𝑑

∆𝜉 = 92.5 𝑚𝑚

𝐼 𝑟 𝑈 0

𝑈 𝑟 2 𝑈 0 ∗ 𝑈 𝑟 (𝐼 𝑟 + 𝐼 0 )

𝑧 = −𝑑

ω 𝑅

𝒅(𝑅, 𝑡) = 𝒅(𝑅) 𝑠𝑖𝑛(𝜔𝑡 + 𝜓 ₀ (𝑅)).

𝛺(𝑅) 𝒅(𝑅)

𝑅 𝛿(𝑅)

𝑅 𝐵

𝛺(𝑅)

𝛺(𝑅) = 2𝜋 𝜆 𝛿(𝑅).

𝑆 𝐵

𝑅

𝑅 1 𝑅 2

𝑑 = 𝑅 ₂ − 𝑅 ₁

𝛿(𝑅) = |𝑆 𝑅 ₁ | + | 𝑅 ₁ 𝐵| − |𝑆 𝑅 ₂ | − | 𝑅 ₂ 𝐵| =

= 𝑠 ₁ 𝑆 𝑅 ₁ + 𝑏 ₁ 𝑅 ₁ 𝐵 − 𝑠 ₂ 𝑆 𝑅 ₂ − 𝑏 ₂ 𝑅 ₂ 𝐵,

𝑠 ₁ 𝑠 ₂ 𝑏 ₁ 𝑏 ₂

𝑆 𝑅 _𝑖 𝑅 _𝑖 𝐵 𝑆

𝑅 _𝑖 𝑅 _𝑖 𝐵

𝒅(𝑅) = 𝑹 _𝟏 𝑩 − 𝑹 _𝟐 𝑩 = 𝑺 𝑹 _𝟐 − 𝑺 𝑹 _𝟏 .

𝑑 |𝑆 𝑅 𝑖 | | 𝑅 _𝑖 𝐵|

s ₁ , s ₂

𝑏 ₁ , 𝑏 ₂ 𝑏 = 𝑏 ₁ = 𝑏 ₂ 𝑠 = 𝑠 ₁ = 𝑠 ₂

𝛿(𝑅) = 𝒅(𝑅)[𝒃(𝑅) − 𝒔(𝑅)].

𝒆(𝑅) = 2𝜋

𝜆 [𝒃(𝑅) − 𝒔(𝑅)].

𝑑

𝛺(𝑅) = 𝒅(𝑅)𝒆(𝑅).

𝑡 𝑅

𝑈 _𝑂 (𝑅, 𝑡) 𝑈 ₀ (𝑅)

𝑈 _𝑂 (𝑅, 𝑡) = 𝑈 ₀ (𝑅)𝑒𝑥 𝑝(𝑗𝛺(𝑅) 𝑠𝑖𝑛[𝜔𝑡 + 𝜓 ₀ (𝑅)]).

ℎ ≈ |𝑈 _𝑜 + 𝑈 _𝑟 | ² 𝑈 _𝑟

𝑇

ℎ(𝜉, 𝜂, 0) = ∫ ℎ(𝜉, 𝜂, 0, 𝑡)

𝑇 0

𝑑𝑡.

𝑈 _{𝑟𝑒𝑎𝑙} (𝑅, 𝑡) ≈ ∫ 𝑈 ₀ (𝑅)𝑒𝑥𝑝(𝑗𝛺(𝑅) 𝑠𝑖𝑛[𝜔𝑡 + 𝜓 ₀ (𝑅)])

𝑇 0

𝑑𝑡.

∑ 𝐽 _𝑛 (𝐴) 𝑒𝑥𝑝(𝑗𝑛𝐵)

∞

𝑛=0

= 𝑒𝑥𝑝(𝑗𝐴 𝑠𝑖𝑛(𝐵)),

𝐽 𝑛

𝑈 _{𝑟𝑒𝑎𝑙} (𝑅, 𝑡) ≈ ∫ ∑ 𝑈 ₀ (𝑅)𝐽 _𝑛 (𝛺(𝑅)) 𝑒𝑥𝑝(𝑗𝑛[𝜔𝑡 + 𝜓 ₀ (𝑅)])

∞

𝑛=0 𝑇 0

𝑑𝑡 =

= ∑ 𝑈 ₀ (𝑅)𝐽 _𝑛 (𝛺(𝑅)) ∫ 𝑒𝑥𝑝(𝑗𝑛[𝜔𝑡 + 𝜓 ₀ (𝑅)]) 𝑑𝑡

𝑇

0

∞

𝑛=0

.

1

𝑇 ≫ 2𝜋 /𝜔

𝑈 _{𝑅𝑒𝑎𝑙} (𝑅) = 𝑙𝑖𝑚

𝑇→∞ ∑ 𝑈 ₀ (𝑅)𝐽 _𝑛 (𝛺(𝑅)) ∫ 𝑒𝑥𝑝(𝑗𝑛 [𝜔𝑡 + 𝜓 ₀ (𝑅)]) 𝑑𝑡 =

𝑇

0

∞

𝑛=0

= 𝑈 ₀ (𝑅)𝐽 ₀ (𝛺(𝑅)).

ψ 0 (𝑅)

𝐼(𝑅) = |𝑈(𝑅)| ² = 𝐼 ₀ (𝑅)𝐽 ₀ ² (𝛺(𝑅)),

𝐼 0

𝑅 𝐽 0 2

𝐽 ₀ ² (0) = 1

𝐽 ₀ ² (𝛺(𝑅)) = 0

1

T ≈ 1 s

2π

ω ≈ 0.01 s

2

𝑈 ₀ 𝑈

(𝑅) = 𝑈(𝑅)𝑈

(𝑅)/|𝑈

|

𝒅(𝑅) = (0, 0, 𝑑 _𝑍 (𝑅)) 𝒔(𝑅) = (0,0, −1) 𝒃(𝑅) = (0,0,1)

𝒆(𝑅) = 2𝜋/𝜆(𝒃(𝑅) − 𝒔(𝑅)) = (0,0,4𝜋/𝜆)

𝑑 𝑍 = 𝑏 𝑚 𝜆/4𝜋 𝑏 𝑚 𝐽 0





𝐽 0 2



𝐽 ₀ ²

𝑑 _𝑍 (𝑅) = 𝑏 _𝑚 𝜆 4𝜋 ⁄ 𝑏 _𝑚 𝐽 ₀

𝑏 _𝑚



𝐽 _𝑛 (𝛺) = ∑ (−1) ^𝑖 𝛺 ^𝑛+2𝑖 /(2 ^𝑛+2𝑖 𝑖! (𝑛 + 𝑖)!)

∞ 𝑖=0

|Ω| < ∞ 𝛺

𝛺 → 0

𝛺→0 𝑙𝑖𝑚

𝑑𝐽 ₀

𝑑𝛺 = 0.

𝐼 = 𝐽 ₀ ² (𝛺 → ∞ ) ≈ 𝑐𝑜𝑠 ² (𝛺 − 𝜋 4 ⁄ )

𝛺 = (4𝜋 𝜆 ⁄ )𝑑 _𝑍

∆ 𝛺 = 𝜋 ∆ 𝑑 _𝑍 = 𝜆/4

𝜆/8

1

𝐼 ≈ 𝑐𝑜𝑠 ² (4𝜋𝑑 _𝑍 ⁄ − 𝜋 4 𝜆 ⁄ )

2

𝑠 = ^𝜆𝑑𝜋

2𝑁∆𝜉

𝑠 = ^𝜋

2 ∆𝑥~2∆𝑥

𝜆/16

5 𝜇𝑚

𝑓 ₀ 𝐵 𝑓 _𝑚𝑜𝑑

𝑣

Λ = 𝑣 𝑓 ⁄

𝜆 = 𝑐 𝑓 ⁄ ₀ 𝜃 _𝐵𝐴

𝑠𝑖𝑛𝜃 _𝐵𝐴 = 𝜆 2𝛬

𝑓 ₁ = 𝑓 ₀ (1 + 𝑣

𝑐 ) 𝑠𝑖𝑛𝜃 _𝐵𝐴 ≈ 𝑓 ₀ + 𝑓 _𝑚𝑜𝑑 .

𝑓

𝑢(𝑡) = 𝐴(𝑡)𝑠𝑖𝑛 (2𝜋𝑓 _𝑚𝑜𝑑 𝑡 − 𝜙(𝑡)),

𝐴(𝑡) 𝜙(𝑡)

𝑣

𝛥𝑛(𝑦, 𝑡) = 𝐶𝑢(𝑡 − 𝜏),

𝐶 𝜏 = 𝑡 ₀ − 𝑦 𝑣 ⁄

𝑡 ₀ = 𝐿/2𝑣

𝑈 ₀ (𝑦, 𝑡) = 𝑈 _𝑖𝑛 𝑒𝑥𝑝 (𝑗 2𝜋𝑑𝐶

𝜆 𝐴(𝑡 − 𝜏)𝑠𝑖𝑛[2𝜋𝑓 _𝑚𝑜𝑑 (𝑡 − 𝜏) − 𝜙(𝑡 − 𝜏)]),

𝑈 _𝑖𝑛

𝑈 ₊₁ (𝑦, 𝑡) = 𝜋𝑑𝐶

𝜆 𝑈 _𝑖𝑛 𝐴(𝑡 − 𝜏)𝑒𝑥𝑝(−𝑗𝜙(𝑡 − 𝜏)) ×

× 𝑒𝑥𝑝 (𝑗 2𝜋𝑦

𝛬 ) 𝑒𝑥𝑝(𝑗2𝜋𝑓 _𝑚𝑜𝑑 (𝑡 − 𝑡 ₀ )).

𝑈 𝑜 𝑈 𝑟

𝑈 _𝐵𝐶𝑜 , 𝑈 _𝐵𝐶𝑟

ℎ ≈ |𝑈 _𝑜 𝑈 _𝐵𝐶𝑜 + 𝑈 _𝑟 𝑈 _𝐵𝐶𝑟 | ² =

= |𝑈 _𝑜 𝑈 _𝐵𝐶𝑜 | ² + |𝑈 _𝑟 𝑈 _𝐵𝐶𝑟 | ² + 𝑈 _𝑜 𝑈 _𝐵𝐶𝑜 𝑈 _𝑟 ^∗ 𝑈 _𝐵𝐶𝑟 ^∗ + 𝑈 _𝑜 ^∗ 𝑈 _𝐵𝐶𝑜 ^∗ 𝑈 _𝑟 𝑈 _𝐵𝐶𝑟 . 𝐴, 𝜙, 𝑓 _𝑚𝑜𝑑

𝐼 _𝑟 𝑈 ₀

𝜗

𝑈 _𝑟 = 𝑒𝑥𝑝(−𝑗𝑘𝑥 𝑠𝑖𝑛(𝜗))

𝜗 = 0

𝒰 _𝓇 , 𝒰 _ℴ , ℋ 𝑈 _𝑟 , 𝑈 _𝑜 ℎ

ℋ ≈ (𝒰 _𝓇 + 𝒰 _ℴ ) ∗ (𝒰 _𝓇 ^∗ + 𝒰 _ℴ ^∗ ) =

= 𝒰 _𝓇 ⋆ 𝒰 _𝓇 + 𝒰 _ℴ ⋆ 𝒰 _ℴ + 𝒰 _ℴ ∗ 𝒰 _𝓇 ^∗ + 𝒰 _ℴ ^∗ ∗ 𝒰 _𝓇 ,

∗,⋆



𝒰 _𝓇 = ℱ{𝑈 _𝑟 } = 𝛿(𝑓 − 𝑓 _𝑐 )

𝑓 𝑐 = ^{𝑠𝑖𝑛(𝜗)}

𝜆

𝛿 ∗ 𝑔 = 𝑔 ∗ 𝛿 = 𝑔 𝑔

ℋ ≈ 𝛿 ⋆ 𝛿 + 𝒰 _ℴ ⋆ 𝒰 _ℴ + 𝒰 _ℴ ∗ 𝛿 + 𝛿 ∗ 𝒰 _ℴ ^∗ = 𝛿 + 𝒰 _ℴ ⋆ 𝒰 _ℴ + 𝒰 _ℴ + 𝒰 _ℴ ^∗ .

𝑓 _𝐵 𝑑 ₀

𝑑 𝑓 _𝐵 = ^𝑑

𝑑𝜆 𝒰 _ℴ

𝒰 _ℴ ^∗

𝑓 _𝑐

𝜗 𝜗

𝑓 _𝑐 < 𝑓 _𝑚𝑎𝑥 = ¹

2∆𝜉

𝑓 _𝑐 > 1.5𝑓 _𝐵 = 1.5 ^𝑑

𝑑𝜆

𝒰 _ℴ

〈−𝑓 _{𝑚𝑎𝑥,} 𝑓 _𝑚𝑎𝑥 〉

𝑓 𝑐 = 0

𝑈 𝑟 = |𝑈 𝑟 | 𝑈 𝑜 = |𝑈 𝑜 |𝑒𝑥𝑝(−𝑗𝜑 _𝑜 )

ℎ ≈ |𝑈 _𝑟 | ² + |𝑈 _𝑜 | ² + 2|𝑈 _𝑟 ||𝑈 _𝑜 | 𝑐𝑜𝑠(𝜑 _𝑜 )

ℎ ≈ 𝑎 + 𝑏 𝑐𝑜𝑠(𝜑 _𝑜 ),

𝑎 = |𝑈 𝑟 | ² + |𝑈 𝑜 | ² 𝑏 = 2|𝑈 _𝑟 ||𝑈 _𝑜 |

𝑈 ₀ = |𝑈 _𝑜 |exp (−𝑗𝜑 _𝑜 )

|𝑈 _𝑜 | = √𝐼 _𝑜

|𝑈 _𝑜 |

𝜑 _𝑜

∆𝜑

𝑁

ℎ _𝑖 ≈ 𝑎 + 𝑏 𝑐𝑜𝑠(𝜑 _𝑜 + ∆𝜑 _𝑖 ),

𝑖 𝑖 = 1,2,3, … 𝑁 𝜑 𝑜

𝜑 _𝑜 (𝜉, 𝜂) ∆𝜑 _𝑖 (𝑡)

∆𝜑 _𝑖

𝜑 _𝑜

𝑓 𝑅−𝑚𝑜𝑑 𝑓 𝑂−𝑚𝑜𝑑

𝜔 _𝑟 = 2𝜋(𝑓 ₀ + 𝑓 _{𝑅−𝑚𝑜𝑑} ) 𝜔 _𝑂 = 2𝜋(𝑓 ₀ +

𝑓 _{𝑂−𝑚𝑜𝑑} ) 𝑓 ₀ 𝜔 _𝑅 ≠ 𝜔 _𝑂

ℎ ≈ 𝑎 + 𝑏 𝑐𝑜𝑠(𝜑 _𝑜 + (𝜔 _𝑂 − 𝜔 _𝑅 )𝑡).

𝜔 _𝐵 = 2(𝜔 _𝑂 − 𝜔 _𝑅 ) = 2∆𝜔

∆𝜑 𝑖

𝐹𝑃𝑆

∆𝜑

∆𝜑(∆𝜔) = 𝜑(𝑡) − 𝜑 (𝑡 + 1

𝐹𝑃𝑆 ) = (𝜔 _𝑂 − 𝜔 _𝑅 ) 1

𝐹𝑃𝑆 = ∆𝜔 𝐹𝑃𝑆 .

∆𝜑 ℎ _𝑖 ℎ _𝑖+1

𝐹𝑃𝑆 = 6.5 𝐻𝑧 ∆𝜔

∆𝜔

∆𝜑 = 𝜋/2

𝐹𝑃𝑆 = 6.5 𝐻𝑧

∆𝑓 = 1.625𝐻𝑧

∆𝜑(∆ω) 𝑖 ∆𝜑 𝑖

∆𝜑 _𝑖 = 0, 𝜋/2, 𝜋, 3𝜋 /2 𝑖 = 1,2,3,4

ℎ ₁ ≈ 𝑎 + 𝑏 𝑐𝑜𝑠(𝜑 _𝑜 ),

ℎ ₂ ≈ 𝑎 + 𝑏 𝑐𝑜𝑠 (𝜑 _𝑜 + 𝜋

2 ) = 𝑎 − 𝑏 𝑠𝑖𝑛(𝜑 _𝑜 ), ℎ ₃ ≈ 𝑎 + 𝑏 𝑐𝑜𝑠(𝜑 _𝑜 + 𝜋) = 𝑎 − 𝑏 𝑐𝑜𝑠(𝜑 _𝑜 ),

ℎ ₄ ≈ 𝑎 + 𝑏 𝑐𝑜𝑠 (𝜑 _𝑜 + 3𝜋

2 ) = 𝑎 + 𝑏 𝑠𝑖𝑛(𝜑 _𝑜 ).

𝑎, 𝑏, 𝜑 _𝑜 𝑎

ℎ ₁ − ℎ ₃ = 2𝑏𝑐𝑜𝑠(𝜑 _𝑜 ),

ℎ ₄ − ℎ ₂ = 2𝑏𝑠𝑖𝑛(𝜑 _𝑜 ).

𝑈 _𝑜 = |𝑈 _𝑜 | exp(−𝑗𝜑 _𝑜 ) = |𝑈 _𝑜 |(cos(𝜑 _𝑜 ) − 𝑗𝑠𝑖𝑛(𝜑 _𝑜 ))

𝑈 _𝑜 = 𝐴((ℎ ₁ − ℎ ₃ ) − 𝑗(ℎ ₄ − ℎ ₂ )),

𝐴 𝐴 ≈ 1/4|𝑈 _𝑟 |

ℎ

𝑈 ₀

𝑑 _0−𝑥 = 49 𝑚𝑚 𝑑 _0−𝑦 = 21 𝑚𝑚

𝑑 = 600 𝑚𝑚

𝑁 = 𝑀 = 2048 𝑝𝑖𝑥 ∆𝜉 = ∆𝜂 = 3.45 𝜇𝑚

𝜆 = 532 𝑛𝑚

𝒰 _ℴ

𝑓 𝑐−𝜂

𝑓 _𝐵−𝜂 = ^𝑑

𝑑𝜆 = 66 𝑚𝑚 ⁻¹

𝒰 _ℴ ⋆ 𝒰 _ℴ 𝑓 _𝑐−𝜂 < 1.5𝑓 _𝐵

𝜔 𝑜 = 2𝜋 𝑓 𝑂 𝑓 𝑂 = 40𝑀𝐻𝑧

𝑓 𝑟−𝑚𝑜𝑑 = 𝐹𝑃𝑆 4 ⁄ = 1.625𝐻𝑧

∆𝜑 = 𝜋/2

𝜔 _𝑟 = 2𝜋(𝑓 ₀ + 𝑓 _{𝑟−𝑚𝑜𝑑} )

𝜂 𝑁 𝒰

𝜂 = ∑ |𝒰 _𝑓

_ℴ (𝑓 _𝑏 )| ² ⁄ 𝑁 _𝒰

𝑑 = 350 𝑚𝑚

𝑓 𝐵−𝜂 = 121𝑚𝑚 ⁻¹

𝑓 _𝐵−𝜉 = 263 𝑚𝑚 ⁻¹ 𝑓 _𝑚𝑎𝑥 = 290 𝑚𝑚 ⁻¹

𝑑 = 350 𝑚𝑚

1

𝜂

𝑓 0 + 𝑓 𝑅−𝑚𝑜𝑑

𝑈 _𝐵𝐶𝑟 = 𝑒𝑥𝑝(𝑗2𝜋(𝑓 ₀ + 𝑓 _{𝑅−𝑚𝑜𝑑} )𝑡) = 𝑒𝑥𝑝 (𝑗(𝜔 ₀ + 𝜔 _{𝑅−𝑚𝑜𝑑} )𝑡)

𝜔 𝑅−𝑚𝑜𝑑 = 𝑚𝜔 𝜔

𝜔 ₀ = 2𝜋𝑓 ₀ : 𝑈 _𝐵𝐶𝑜 = 𝑒𝑥𝑝 (𝑗𝜔 ₀ 𝑡)

𝑈 _{𝑟𝑒𝑎𝑙} (𝑅, 𝑡) ≈ ∫ 𝑈 _𝑜 𝑈 _𝐵𝐶𝑜 𝑈 _𝑟 ^∗ 𝑈 _𝐵𝐶𝑟 ^∗

𝑇 0

𝑑𝑡 =

= ∫ 𝑒𝑥𝑝(𝑗𝛺(𝑅) 𝑠𝑖𝑛[𝜔𝑡 + 𝜓 ₀ (𝑅)])

𝑇 0

𝑒𝑥𝑝(−𝑗𝑚𝜔𝑡) 𝑑𝑡,

𝑈 𝑟 ∗ = 1

𝑈 _{𝑟𝑒𝑎𝑙} (𝑅, 𝑡) ≈ 𝑙𝑖𝑚

𝑇→∞ ∑ 𝐽 _𝑛 (𝛺(𝑅)) ∫ 𝑒𝑥𝑝(𝑗𝑛[𝜔𝑡 + 𝜓 ₀ (𝑅)]) ×

𝑇

0

∞

𝑛=1

× 𝑒𝑥𝑝(−𝑗𝑚𝜔𝑡) 𝑑𝑡.

𝑛 = 𝑚

|𝑈(𝑅)| ≈ |𝐽 _𝑚 (𝛺(𝑅))|.

𝛺(𝑅) ≪ 1

|𝑈| ≈ |𝐽 ₀ (0)|

|𝑈| ≈ |𝐽 ₁ (0)|

lim Ω→0

𝑑𝐽

𝑑Ω = 0

𝑙𝑖𝑚 𝛺→0

𝑑𝐽 ₁ 𝑑𝛺 = 0.5

𝑓 = 𝜔 2𝜋 ⁄ = 6000 𝐻𝑧

𝑢

𝑑 𝑍 = 𝐶 𝑣 𝑢 𝐶 𝑣

𝑓 𝑜 = 𝑓 𝑟 = 𝑓 0 = 40 𝑀𝐻𝑧

𝑓 ₀

|𝑈 _𝐽0 | ≈ |𝐽 ₀ ( ^4π

𝜆 𝐶 _𝑣 𝑢)| 𝒆 =

(0, 0, 4𝜋 𝜆) ⁄

𝑢 = 0.05 𝑉

Δ𝑢 = 0.05 𝑉

𝑢 = 0.5 𝑉

𝑓 _𝑟 = 𝑓 ₀ + 𝑓 _{𝑅−𝑚𝑜𝑑} =

𝑓 ₀ + 𝑚𝑓 𝑚 = 1 𝑓 _𝑟 = 40.006 𝑀𝐻𝑧

𝑓 _𝑜 = 𝑓 ₀ = 40 𝑀𝐻𝑧

|𝑈 𝐽1 | ≈ |𝐽 1 ( ^4π

𝜆 𝐶 𝑣 𝑢)|

|𝑈 𝐽0 |

|𝑈 _𝐽1 |

|𝐽 ₀ ( ^4𝜋

𝜆 𝐶 _𝑣 𝑢)|

|𝐽 ₁ ( ^4𝜋

𝜆 𝐶 _𝑣 𝑢)|

𝐶 𝑣

|𝑈 _𝐽0 | |𝑈 _𝐽1 |

𝐶 _𝑣 = 589 |𝑈 _𝐽0 | 𝐶 _𝑣 = 585 |𝑈 _𝐽1 |

𝑑 _{𝑍−𝑛𝑜𝑛} = 294 .5 𝑛𝑚 𝑑 _{𝑍−𝑚𝑜𝑑} = 292 .5 𝑛𝑚

𝑏 ₂

𝐶 _𝑣 =

589 𝑑 _𝑍 = 235.6 𝑛𝑚

𝐶 _𝑣 = 141 𝐶 _𝑣 = 136

|𝑈 _𝐽0 | |𝑈

|

𝑑 𝑍−𝑛𝑜𝑛 (𝑢 = 0.5) = 70.5 𝑛𝑚 𝑑 𝑍−𝑚𝑜𝑑 (0.5) = 68 𝑛𝑚

𝛺 = ^4𝜋

𝜆 𝑑 𝑧

𝐽 1−𝑚𝑎𝑥 (𝛺 = 1.84) = 0.58 𝛺 < 0.5

𝐽 1 (𝛺) ≈ ¹

2 𝛺

|𝑈 _𝐽1 |~0.001

𝑑 _𝑧 = 2 𝜆 4𝜋 |𝑈 _𝐽1 |,

𝑑 𝑧 ~0.085 𝑛𝑚 𝑑 𝑧 ~ ^𝜆

6000

〈 ^|𝑈

^|

〈|𝑈

|〉 〉 |𝑈 _𝐽1 |

〈|𝑈 _𝐽1 |〉

〈|𝑈 _𝐽1 |〉

|𝑈 𝐽1 |

𝑝(𝐼) = 𝑒𝑥𝑝 (− _〈𝐼〉 ^𝐼 )

𝐼 〈𝐼〉

𝑑 _𝑧 ~0.42 𝑛𝑚

|𝑈 _𝐽1 (𝑅)| = |𝑈 ₀ (𝑅)||𝐽 ₁ (𝛺(𝑅))|,

|𝑈 ₀ (𝑅)|

𝑆

𝑆(𝑅) = 𝑑|𝑈 _𝐽1 (𝑅)|

𝑑𝑑 _𝑍 =

𝑑 (|𝑈 ₀ (𝑅)| |𝐽 ₁ ( 4𝜋

𝜆 𝑑 _𝑍 (𝑅))|) 𝑑𝑑 _𝑍

𝐽 ^′ ₁ (Ω) = 1/2[𝐽 ₀ (Ω) − 𝐽 ₂ (Ω)]

𝑆(𝑅) = 2𝜋

𝜆 |𝑈 ₀ (𝑅)| (|𝐽 ₀ ( 4𝜋

𝜆 𝑑 _𝑍 (𝑅))| − |𝐽 ₂ ( 4𝜋

𝜆 𝑑 _𝑍 (𝑅))|).

|𝑈 ₀ (𝑅)|

𝐽 ₀ (Ω(𝑅)) ≈ 1 𝐽 ₂ (Ω(𝑅)) ≈ 0

𝑆(𝑅) = 𝑑|𝑈 _𝐽1 (𝑅)|

𝑑𝑑 _𝑍 ≈ 2𝜋

𝜆 |𝑈 ₀ (𝑅)|,

|𝑈 ₀ (𝑅)| 𝑅

|𝑈 ₀ (𝑅)| |𝑈 ₀ | 𝑅1 |𝑈 ₀ (𝑅1)| = 1

|𝑈 ₀ (𝑅2)| = 0.25 𝑅2

|𝑈

(𝑅)|

𝑅1

|𝑈 _𝐽1 (𝑅1)|~0.001

∆|𝑈 _𝐽1 |

𝑑 _{𝑍−𝑀𝐼𝑁} (𝑅2) = ^𝜆

2𝜋

∆|𝑈

|

|𝑈

(𝑅2)| = ^𝜆

2𝜋

∆|𝑈

| 0.25 = 5 ^𝜆

2𝜋 ∆|𝑈 _𝐽1 |

|𝑈 _𝐽0 |

|𝑈

(𝑅)|

𝑓 _𝑟 = 𝑓 ₀ + 𝑚𝑓 =

40𝑀𝐻𝑧 + 50 ∗ 100𝐻𝑧 |𝑈 _𝐽50 |

|𝑈 𝐽50 |

|𝑈

| |𝑈 _𝐽50 | |𝑈 _𝐽150 |

𝑓 ₀ ¹

𝑢(𝑡) = 𝑠𝑖𝑛 (2𝜋𝑓 ₀ 𝑡 − 𝜙(𝑡))

𝑓

= 40 𝑀𝐻𝑧

𝑈 ₊₁ = 𝑈 _𝑖𝑛 𝑒𝑥𝑝(−𝑗𝜙)𝑒𝑥𝑝(𝑗2𝜋𝑓 ₀ )).

𝜔 𝜙 𝐵𝐶

𝑈 _𝑟 𝑈 _𝐵𝐶𝑟 = 𝑒𝑥𝑝(𝑗𝜙 _𝐵𝐶 𝑠𝑖𝑛(𝜔𝑡))𝑒𝑥𝑝(𝑗2𝜋𝑓 ₀ ),

𝑈 _𝑜 𝑈 _𝐵𝐶𝑜 = 𝑒𝑥𝑝(𝑗𝛺 𝑠𝑖𝑛[𝜔𝑡 + 𝜓 ₀ ]) 𝑒𝑥𝑝(𝑗2𝜋𝑓 ₀ ).

𝑈 _{𝑟𝑒𝑎𝑙} ≈ ∫ 𝑈 ₀ ^𝑇 𝑜 𝑈 _𝐵𝐶𝑜 𝑈 _𝑟 ^∗ 𝑈 _𝐵𝐶𝑟 ^∗ 𝑑𝑡 = ∫ 𝑒𝑥𝑝 (𝑗(𝛺 𝑠𝑖𝑛[𝜔𝑡 + 𝜓 ₀ ^𝑇 0 ] − 𝜙 _𝐵𝐶 𝑠𝑖𝑛(𝜔𝑡))) 𝑑𝑡 = 𝐽 ₀ (√𝛺 ² − 2𝛺𝜙 _𝐵𝐶 𝑐𝑜𝑠 𝜓 ₀ + 𝜙 _𝐵𝐶 ² )

𝜓 ₀ = 0

|𝑈 _{𝑟𝑒𝑎𝑙} | ≈ |𝐽 ₀ (𝛺 − 𝜙 _𝐵𝐶 )|.

𝛺 = 𝜙 _𝐵𝐶

𝜙

𝜙 _𝐵𝐶 = 0 𝑟𝑎𝑑

𝜋/31 2𝜋

|𝐽 ₀ (𝛺 − 𝜙 _𝐵𝐶 )|

〈0, 𝜋〉

𝑐𝑜𝑠(𝑥 − 𝑦) = 𝑐𝑜𝑠(𝑥)𝑐𝑜𝑠(𝑦) + 𝑠𝑖𝑛(𝑥)𝑠𝑖𝑛(𝑦)

𝛺

𝛺 ^∗

𝜙 𝐵𝐶

|𝑈 _𝑖 | = 𝑎 + 𝑏|𝐽 ₀ (𝛺 − 𝜙 _𝐵𝐶𝑖 )|~𝑎 + 𝑏|𝑐𝑜𝑠(𝛺 ^∗ − 𝜙 _𝐵𝐶𝑖 )|,

𝑎 𝑏

𝜙 _𝐵𝐶𝑖 = 𝜋/4

|𝑈 ₁ |~𝑎 + 𝑏|𝑐𝑜𝑠(𝛺 ^∗ )|,

|𝑈 ₂ |~𝑎 + 𝑏|𝑐𝑜𝑠(𝛺 ^∗ + 𝜋/4)| = 𝑎 + 𝑏|𝑠𝑖𝑛(𝛺 ^∗ )|,

|𝑈 ₃ |~𝑎 + 𝑏|𝑐𝑜𝑠(𝛺 ^∗ + 𝜋/2)| = 𝑎 − 𝑏|𝑐𝑜𝑠(𝛺 ^∗ )|,

|𝑈 ₄ |~𝑎 + 𝑏|𝑐𝑜𝑠(𝛺 ^∗ + 3𝜋/4)| = 𝑎 − 𝑏|𝑠𝑖𝑛(𝛺 ^∗ )|.

𝑎, 𝑏, 𝛺 𝑎

|𝑈 ₁ | − |𝑈 ₃ | = 2𝑏|𝑐𝑜𝑠(𝛺 ^∗ )|,

|𝑈 ₄ | − |𝑈 ₂ | = 2𝑏|𝑠𝑖𝑛(𝛺 ^∗ )|,

𝛺 ^∗

𝛺 ^∗ = 𝑎𝑡𝑎𝑛 ( |𝑈 ₄ | − |𝑈 ₂ |

|𝑈 ₁ | − |𝑈 ₃ | ).

𝜋/4

𝛺 ^∗ = 𝑎𝑡𝑎𝑛 ( −4𝐴 − 12𝐵 + 16𝐶 + 24|𝑈 ₇ |

3𝐷 + 4𝐴 − 12𝐸 − 21𝐹 − 16𝐺 ),

𝐴 = |𝑈 2 | − |𝑈 ₁₂ |, 𝐵 = |𝑈 ₃ | + |𝑈 ₁₁ | + |𝑈 ₄ | + |𝑈 ₁₀ |, 𝐶 = |𝑈 ₆ | + |𝑈 ₈ |, 𝐷 = |𝑈 ₁ | − |𝑈 ₁₃ |, 𝐸 =

|𝑈 ₄ | − |𝑈 ₁₀ | 𝐹 = |𝑈 ₅ | − |𝑈 ₉ | 𝐺 = |𝑈 ₆ | − |𝑈 ₈ | 𝜋/8

|𝑐𝑜𝑠 𝑥| |𝐽 ₀ (𝑥)|

𝛺 ^∗ 〈−𝜋, 𝜋〉

2𝜋 𝛺 ^∗

𝛺

∆= 𝛺 − 𝛺 ^∗ |𝑐𝑜𝑠 𝑥| |𝐽 ₀ (𝑥)|

∆

𝛺 = 𝛺 ^∗ + ∆

〈−𝝅, 𝝅〉

〈−𝜋/2, 𝜋/2〉

Ω ^∗ Ω

𝛺

|𝑈 _𝐶𝑎𝑙 | = |𝐽 ₀ (Ω)|

|𝑈 _{𝑀𝑒𝑎𝑠} |

𝛺

𝑑 _𝑧 = ^𝜆

4𝜋 𝛺

𝑑 _𝑧 (𝑅)

̅̅̅̅̅̅̅̅ = ¹

𝑁 ∑ ^𝑁 _𝑛=1 𝑑 _{𝑧 𝑛} (𝑅)

𝜎(𝑅) = √ ¹

𝑁 ∑ ^𝑁 _𝑛=1 (𝑑 _{𝑧 𝑛} (𝑅) − 𝑑 ̅̅̅̅̅̅̅̅) _𝑧 (𝑅) ²

𝜎(𝑅) = Α + Β 𝑑 ̅̅̅̅̅̅̅̅ 𝑧 (𝑅)

𝜎(𝑅) = 0.05 + 0.01 𝑑 ̅̅̅̅̅̅̅̅ [𝑛𝑚], _𝑧 (𝑅)

100𝜎 𝑑

̅̅̅̅ = ⁵

𝑑

̅̅̅̅ + 1 [%]

𝑑 𝑧

̅̅̅̅ = 300 𝑛𝑚 𝜎 = 3. 05 𝑛𝑚 ~ 1.17 %

σ

𝜎

3𝜎 3𝜎 = 9.2 𝑛𝑚~3.5% 𝑑 ̅̅̅̅ = 300 𝑛𝑚 _𝑧





𝜆/6000

20𝜆 120000



1

20𝜆

𝑢 _𝐵𝐶𝑅 (𝑡) = 𝑠𝑖𝑛 (2𝜋𝑓 _{𝑅−𝑚𝑜𝑑} 𝑡 − 𝜙(𝑡)) 𝑢 _𝐵𝐶𝑂 (𝑡) = 𝑠𝑖𝑛 (2𝜋𝑓 _{𝑂−𝑚𝑜𝑑} 𝑡)

𝑈 _𝐵𝐶𝑟 = 𝑒𝑥𝑝(−𝑗𝜙)𝑒𝑥𝑝(𝑗2𝜋𝑓 _{𝑅−𝑚𝑜𝑑} ))

𝑈 _𝐵𝐶𝑜 = 𝑒𝑥𝑝(𝑗2𝜋𝑓 _{𝑅−𝑚𝑜𝑑} ))

ω

𝜙 _𝐵𝐶 : 𝜙 = 𝜙 _𝐵𝐶 sin (𝜔𝑡)

𝑚 𝑓

𝐹𝑃𝑆

𝜋/2 𝑓 𝑅−𝑚𝑜𝑑 = 𝑓 0 + 𝑚𝑓 + 𝐹𝑃𝑆 4 ⁄

𝑈 𝐵𝐶𝑜

𝑒𝑥𝑝 (𝑗𝛺(𝑅) 𝑠𝑖𝑛[𝜔𝑡 + 𝜓 ₀ (𝑅)])

𝑈 _𝑜 𝑈 _𝐵𝐶𝑜 = 𝑈 ₀ 𝑒𝑥𝑝 (𝑗𝛺(𝑅) 𝑠𝑖𝑛[𝜔𝑡 + 𝜓 ₀ (𝑅)])𝑒𝑥𝑝(𝑗2𝜋𝑓 _{𝑅−𝑚𝑜𝑑} ))

𝑈 _𝑜 ℎ

𝑈 _{𝑟𝑒𝑎𝑙} ≈ 𝑈 ₀ ∑ ^∞ _𝑛=−∞ 𝐽 _𝑛 (𝛺 − 𝜙 _𝐵𝐶 ) ∫ _−𝑇/2 ^𝑇/2 𝑒𝑥𝑝(𝑗𝑛[𝜔𝑡 + 𝜓 ₀ ]) × × 𝑒𝑥𝑝(−𝑗𝑚𝜔𝑡) 𝑑𝑡

|𝑈 _{𝑟𝑒𝑎𝑙} | ≈ |𝑈 ₀ ||𝐽 _𝑛 (𝛺 − 𝜙 _𝐵𝐶 )|

1

𝐹𝑃𝑆 𝑀 ⁄ 𝑀

1000 𝐻𝑧 200 𝜇𝑉 200 𝑚𝑉

𝑖 = 0,1,2 … 12 𝜙 _𝐵𝐶 = 𝑖𝜋/8

𝑑 _𝑧

0.1 𝑛𝑚

100000

|𝑈 _𝐽𝑛 | = |𝑈 ₀ | (1 + 𝑉 |𝐽 _𝑛 ( ^4𝜋

𝜆 𝑑 _𝑧 − 𝜙 _𝐵𝐶 )|) 𝑛 _𝑆 𝑛 _𝐸∗ + 𝑛 _𝐸+ + 𝑛 _𝐷

|𝑈 ₀ |

𝑉

𝑛 𝑆

𝑛 𝐸+ 𝑛 𝐸∗

𝑛 _𝐷

𝛺 _𝐷

𝐴 𝐵

|𝑈 _𝐽𝑛 | = 𝐴 + 𝐵 |𝐽 _𝑛 ( 4𝜋

𝜆 𝑑 _𝑧 − 𝜙 _𝐵𝐶 )|.

𝑉 = 2|𝑈 _𝑟 ||𝑈 _𝑜 |/(|𝑈 _𝑟 | ² + |𝑈 _𝑜 | ² ) 𝐵 = |𝑈 0 |𝑉𝑛 _𝑆 𝑛 𝐸∗

|𝐽 𝑛 ( ^4π

𝜆 𝑑 𝑧 − 𝜙 𝐵𝐶 )| ≪ 1 𝑉 = 1

|𝑈 _𝑟 | = |𝑈 _𝑜 | 𝐴

𝛺 _𝑆

~𝑈 _𝐽𝑛

~𝑈 ₀ 𝑈 = 𝑅𝑒{𝑈} +

𝑗𝐼𝑚{𝑈}

𝑈 0 𝑈 𝐽𝑛

𝑈 ₀ = 𝐵 ₀ 𝑒𝑥𝑝[𝑗(𝛺 _𝑆 + 𝛺 _𝐷0 )],

𝑈 _𝐽𝑛 = 𝐵 ₁ 𝐽 _𝑛 ( 4𝜋

𝜆 𝑑 _𝑧 − 𝜙 _𝐵𝐶 ) 𝑒𝑥𝑝[𝑗(𝛺 _𝑆 + 𝛺 _𝐷1 )],

𝑈 _𝐽𝑛 𝑈 ₀ |𝑈 ₀ |

〈 𝑈 _𝐽𝑛 𝑈 ₀ ^∗

|𝑈 ₀ | ² 〉 = 𝐵 ₁

𝐵 ₀ 𝐽 _𝑛 ( 4𝜋

𝜆 𝑑 _𝑧 − 𝜙 _𝐵𝐶 ) 𝑒𝑥𝑝[𝑗(𝛺 _𝐷1 − 𝛺 _𝐷0 )].

𝛺 _𝑆 ∆𝛺 _𝐷 = 𝛺 _𝐷1 − 𝛺 _𝐷0

𝑈 ₀ 𝑈 _𝐽𝑛

𝑈 𝐽𝑛 𝑈 0 ∗ /|𝑈 0 | ²

〈 〉

𝑎𝑡𝑎𝑛 ( 𝐼𝑚{𝑈}

𝑅𝑒{𝑈} ) = ∆𝛺 _𝐷 + 𝜋

2 𝑠𝑔𝑛 (𝐽 _𝑛 ( 4𝜋

𝜆 𝑑 _𝑧 − 𝜙 _𝐵𝐶 ) + 1)

|𝑈 _𝐽𝑛 | = 𝐵̃ |𝐽 _𝑛 ( 4𝜋

𝜆 𝑑 _𝑧 − 𝜙 _𝐵𝐶 )|,

𝐵̃ = 𝐵 ₁ /𝐵 ₀

𝑈 _𝐽𝑛 𝑈 ₀ 𝐵 ₁ = 𝐵 ₀

|𝑈 𝐽𝑛 |~|𝑈 𝐽𝑛 𝑈 0 ∗ /|𝑈 0 | ² |

𝐵̃, 𝜆, 𝜙 _𝐵𝐶

𝑑|𝑈 _𝐽𝑛 | = √( 𝜕|𝑈 _𝐽𝑛 |

𝜕𝐵̃ 𝑑𝐵̃)

2

+ ( 𝜕|𝑈 _𝐽𝑛 |

𝜕𝜆 𝑑𝜆)

2

,

𝜕|𝑈 _𝐽𝑛 |

𝜕𝐵̃ = |𝐽 _𝑛 ( 4𝜋

𝜆 𝑑 _𝑧 − 𝜙 _𝐵𝐶 )|,

𝜕|𝑈 _𝐽𝑛 |

𝜕𝜆 = 𝐵̃

2 ||𝐽 _𝑛−1 ( 4𝜋

𝜆 𝑑 _𝑧 − 𝜙 _𝐵𝐶 )| − |𝐽 _𝑛+1 ( 4𝜋

𝜆 𝑑 _𝑧 − 𝜙 _𝐵𝐶 )|| 4𝜋 𝜆 ² 𝑑 _𝑧 .

∆𝜆/𝜆 ²

𝜙 𝐵𝐶 = 0

𝑑𝐵̃

𝜀 𝜙̂ _𝐵𝐶

𝜙 _𝐵𝐶 = 𝜙̂ _𝐵𝐶 + 𝜀.

𝜙 _𝐵𝐶

𝜓 ₀ ≠ 0

𝑑𝐵̃

𝐵̃ 𝐵̃ _𝐴𝑉𝐸

σ(𝐵̃)

𝐵̃ = 𝐵̃ _𝐴𝑉𝐸 + 𝜎(𝐵̃).

𝜀 𝐵̃

|𝑈 ₁ | = (𝐵̃ _𝐴𝑉𝐸 + 𝜎(𝐵̃)) |𝑐𝑜𝑠(𝛺 ^∗ )|,

|𝑈 ₂ | = (𝐵̃ _𝐴𝑉𝐸 − 𝜎(𝐵̃)) |𝑐𝑜𝑠 (𝛺 ^∗ + 𝜋 4 + 𝜀)|,

|𝑈 ₃ | = (𝐵̃ _𝐴𝑉𝐸 + 𝜎(𝐵̃)) |𝑐𝑜𝑠 (𝛺 ^∗ + 𝜋 2 + 𝜀)|,

|𝑈 ₄ | = (𝐵̃ _𝐴𝑉𝐸 − 𝜎(𝐵̃)) 𝑐𝑜𝑠 (𝛺 ^∗ + 3𝜋

4 + 𝜀).

|𝑈 ₁ | ≈ (𝐵̃ _𝐴𝑉𝐸 + 𝜎(𝐵̃)) |𝑐𝑜𝑠(𝛺 ^∗ )|,

|𝑈 ₂ | ≈ − (𝐵̃ _𝐴𝑉𝐸 + 𝜎(𝐵̃)) |𝑠𝑖𝑛(𝛺 ^∗ ) 𝑐𝑜𝑠 𝜀|,

|𝑈 ₃ | ≈ − (𝐵̃ _𝐴𝑉𝐸 + 𝜎(𝐵̃)) |𝑐𝑜𝑠(𝛺 ^∗ )𝑐𝑜𝑠 𝜀|,

|𝑈 ₄ | ≈ (𝐵̃ _𝐴𝑉𝐸 + 𝜎(𝐵̃)) 𝑠𝑖𝑛(𝛺 ^∗ ) 𝑐𝑜𝑠 𝜀.

𝑐𝑜𝑠 𝜀 = 1 − 𝜀 ² /2 |𝑈 ₁ | − |𝑈 ₃ | |𝑈 ₄ | − |𝑈 ₂ |

𝛺 ^∗ = 𝑎𝑡𝑎𝑛 ( 2𝐵̃ _𝐴𝑉𝐸 + 2|𝜎(𝐵̃)|

2𝐵̃ _𝐴𝑉𝐸 − 2|𝜎(𝐵̃)|

𝜀 ² 4 − 𝜀 ²

|𝑈 ₄ | − |𝑈 ₂ |

|𝑈 ₁ | − |𝑈 ₃ | ).

𝑐 _𝑣 = ^|σ(𝐵̃)|

𝐵̃

σ ( 𝐵 ̃) ≪ 𝐵 ̃ _𝐴𝑉𝐸

𝛺 ^∗ = 𝑎𝑡𝑎𝑛 ((1 + 2𝑐 _𝑣 ) (1 + ( 𝜀 2 )

2

) |𝑈 ₄ | − |𝑈 ₂ |

|𝑈 ₁ | − |𝑈 ₃ | ).

∆𝛺 ^∗

𝑑𝛺 ^∗ = √( 𝜕𝛺 ^∗

𝜕𝑐 _𝑣 𝑑𝑐 _𝑣 )

2

+ ( 𝜕𝛺 ^∗

𝜕𝜀 𝑑𝜀)

2

.

1

𝑐𝑜𝑠(𝑥 + 𝑦) = 𝑐𝑜𝑠(𝑥) 𝑐𝑜𝑠(𝑦) − 𝑠𝑖𝑛(𝑥) 𝑠𝑖𝑛 (𝑦) ≈ 𝑐𝑜𝑠(𝑥) 𝑐𝑜𝑠 (𝑦) 𝑦

2

(1 + 2𝑐 _𝑣 )(1 + 𝜀 ² /4) ≈ 1 + 2𝑐 _𝑣 + 𝜀 ² /4

2𝑐 _𝑣 𝑡𝑎𝑛𝛺̇ ^∗ + (𝜀 ² /4) 𝑡𝑎𝑛 𝛺̇ ^∗ + 𝑡𝑎𝑛𝛺̇ ^∗ ≈ 𝑡𝑎𝑛𝛺̇ ^∗

(|𝑈 ₄ | − |𝑈 ₂ |)/(|𝑈 ₁ | − |𝑈 ₃ |) = 𝑡𝑎𝑛𝛺̇ ^∗

𝑑𝛺 ^∗ = √(𝑠𝑖𝑛2𝛺̇ ^∗ 𝑑𝑐 _𝑣 ) ² + (𝑠𝑖𝑛2𝛺̇ ^{∗ 𝜀}

4 𝑑𝜀) ²

𝛺̇ ^∗

𝑑𝑑 _𝑧 = 𝑑𝑑 _𝑧

𝑑𝛺 ^∗ 𝑑𝛺 ^∗ =

= 𝜆

4𝜋 √[(𝑠𝑖𝑛2𝛺̇ ^∗ 𝑑𝑐 _𝑣 ) ² + (𝑠𝑖𝑛2𝛺̇ ^∗ 𝜀 4 𝑑𝜀)

2

+ ( ^{𝜕∆(𝛺 ∗ )}

𝜕𝛺 ^∗ 𝑑𝛺 ^∗ ) ² ]

𝑑 _𝑧 = ^𝜆

4𝜋 [𝛺 ^∗ + ∆(𝛺 ^∗ ) ] ∆(𝛺 ^∗ )

𝑑𝑐 _𝑣 𝑑𝜀

𝜙 𝐵𝐶

𝜋 2

𝜙 _𝐵𝐶 = 𝑎𝑐𝑜𝑠 ( 1 2

|𝑈 ₅ | − |𝑈 ₁ |

|𝑈 ₄ | − |𝑈 ₂ | ).

𝜙

𝑅

𝜙 𝐵𝐶 (𝑅)

𝜋 2

𝜀 = 0.0035 𝑟𝑎𝑑 𝑑𝐵̃

𝑑𝐵̃

𝑈 ₁ = 𝐵 ₀ 𝑒𝑥𝑝[𝑗(𝛺 _𝑆 + 𝛺 _𝐷0 )],

𝑈 ₂ = 𝐵 ₁ 𝑒𝑥𝑝[𝑗(𝛺 _𝑆 + 𝛺 _𝐷1 )].

𝑈 2 𝑈 1

|𝑈 ₁ |

〈 𝑈 ₂ 𝑈 ₁ ^∗

|𝑈 ₁ | ² 〉 = 𝐵 ₁

𝐵 ₀ 𝑒𝑥𝑝[𝑗(𝛺 _𝐷1 − 𝛺 _𝐷0 )].

|𝑈 ₁₂ | = 𝐵̃

𝐵̃ = 𝐵 ₁ /𝐵 ₀ 𝑈 ₁ , 𝑈 ₂

|𝑈 ₁₂ |

𝐵̃

1

2

𝑑

= 0 𝐽

(0) = 1

𝐵̃ 𝐴𝑉𝐸 = 0.977

σ(𝐵̃) = 0. 018 𝑐 _𝑣 = ^|σ(𝐵̃)|

𝐵̃

= 0.018

𝐵̃

𝑐 𝑣 𝜀

𝑑𝛺 ^∗ 𝛺 ^∗

sin2𝛺̇ ^{∗ 𝜀}

4 𝑑𝜀 sin2 𝛺 ^{̇ ∗} 𝑑𝑐 𝑣

𝐜 𝐯 𝛆

𝜀

𝐵̃

𝛺 ^∗

𝛺

(𝜕∆(𝛺 ^∗ )/𝜕𝛺 ^∗ )𝑑𝛺 ^∗ 𝜕∆(𝛺 ^∗ )/𝜕𝛺 ^∗

𝑐 _𝑣

𝑑𝑑 _𝑧 𝐵̃

𝜕∆(𝛺 ^∗ )/𝜕𝛺 ^∗

≠

1

𝐵̃ = 𝐵̃ _𝐴𝑉𝐸 + σ(𝐵̃)

𝐵̃ = 𝐵̃ _𝐴𝑉𝐸 −

σ(𝐵̃) = 0.96 |𝐽 1 ( ^4π

𝜆 𝑑 𝑧 )|

𝐵̃ |𝐽 1 ( ^4π

𝜆 𝑑 𝑧 )| 𝑑|𝑈 𝐽1 |

𝑑𝑑 _𝑧

σ(𝐵̃) → 0

σ(𝐵̃) 𝑐 _𝑣 = ^|σ(𝐵̃)|

𝐵̃

𝑐 _𝑣 → 0

𝐵̃ → 𝐵̃ _𝐴𝑉𝐸

σ(𝜀) ≈ 0.1 𝑟𝑎𝑑

𝑢 _𝜀 = 𝜆 𝜎 ² (𝜀)

32𝜋 𝑠𝑖𝑛 ( 8𝜋 𝜆 𝑑 _𝑧 )

B ̃ 𝐵 ̃

𝐵̃ ∈ 〈0.9955, 1.016〉

σ(𝐵̃) ≈ ^𝐵̃

^−𝐵̃

2√3 = 0.018

𝑢 _𝐵̃ = 𝜆𝜎(𝐵̃)

4𝜋 𝑠𝑖𝑛 ( 8𝜋 𝜆 𝑑 _𝑧 ).

𝑢 _∆ = ^{𝜕∆(𝑑 𝑧 )}

𝜕𝑑 𝑧 𝑑𝑑 _𝑧 .

𝑢 _𝐴 = 0.05 + 0.01 𝑑 _𝑧 .

𝑢 _𝑐 = √𝑢 _𝜀 ² + 𝑢 _𝐵̃ ² + 𝑢 _∆ ² + 𝑢 _𝐴 ²

𝑢 𝑐

1

σ(B ̃)

𝑢 𝑐 k 𝑈 = 𝑘𝑢 𝑐

k

𝑈 = 𝑘𝑢 _𝑐 𝑈 = 2𝑢 _𝑐

𝑑 _𝑧

𝑑 𝑧 = 10 𝑛𝑚 𝑑 𝑧 = 100 𝑛𝑚 𝑑 𝑧 = 1000 𝑛𝑚

𝑢 𝑐 [𝑛𝑚] 0.4 1.3 10.1

𝑑 𝑧 ± 𝑈 (𝑘 = 2) (10.0 ± 0.8) 𝑛𝑚 (100.0 ± 2.6) 𝑛𝑚 (1000.0 ± 20.2) 𝑛𝑚









1





𝜋/2

𝑑 = 𝑈 _𝑂𝑈𝑇 𝑈 _𝑃𝑃

𝜆

√2𝜋 ,

𝑈 𝑂𝑈𝑇 𝑈

𝜆

𝑧 _𝑚

𝑝 𝑣

𝑧 _𝑚 = 𝑝 𝑣 ,

𝑇𝐿 = 20𝑙𝑜𝑔 ₁₀ |1 + 𝑧 _𝑚 2𝑧 _𝑎 |,

𝑧 𝑎

𝑧 _𝑚 𝑧 _𝑚

𝑧 𝑚

𝜔 _𝑜